In alphabetical order:
Here are my reasons.
Circulate is one of the first calls in which dancers learn general purpose rules which work from many formations, and their dance action depends only on themselves and the spots on the floor, and not on the other dancers. The simplicity of a Circulate is in stark contrast to its variety (center, end, lead, trailer, once, 1 1/2, etc.).
Counter Rotate qualifies for reasons similar to Circulate. It is simple, elegant, and yet many dancers never master it. Some find the need to memorize it in a case by case basis. Still others invent various cheats (which don't always work). Callers of old misunderstood it, and did it incorrectly from columns. Yet it remains a fundamental dance action.
Scoot Back introduced using the same hand 3 times in a row, which was an anathema to some. Yet it has stood the test of time, is fun to dance, and can have a certain amount of variety.
Run introduces the idea of a runner and runnee. It also provides a quick way to change formations. It extends into Cross Run (which opens up the whole idea of normal vs. Cross), can be done fractionally, and works well with t-bones. A simple call with a simple rule, and many possibilities.
Tag The Line is a fake-out kind of call. The initial "find the center of your line and turn to face it" requires situational awareness which takes some practice to acquire. It introduces the idea of "tag positions" and the ability to have a call take a parameter (0 Tag, 1/4 Tag, 1/2 Tag, 3/4 Tag, and Full Tag), and creates the idea of Extending The Tag (moving from one tag position to the next). Many other calls use Tag The Line as their base (e.g., Tag Chain Thru, Tag The Top).
Remake, while somewhat of a hodgepodge, and having changed definitions over the years is still great for asking dancers to go 1/4, 1/2, and then 3/4. The formations may change wildly while doing this (consider starting in thars or alamo rings). I would have nominated Swing Thru, but Remake has that and more.
Spin Chain The Gears is my poster child for any of our long pattern calls. It is a long sequence of parts, during which you connect and disconnect with people, form sub formations (e.g., stars, very center two, a wave), and may end up moving across the whole square. It is also dancing -- no stop and go here. Any of the other calls like Relay The Deucey, Spin Chain And Exchange The Gears, Relay The Top, Spin Chain The Star would have worked fine.
Chain Reaction is a cleverly named call in which dancers have to follow the rules (the definition) and consider what they should be doing next as each part unfolds (i.e., re-evaluate). "Flow" dancers never get it. Swing Thru is a MS example of this (Facing couples, right-hand waves, left-hand waves, grand), except that Chain Reaction carries this idea of following the rules much further.
The idea of working with Phantoms creates an important step from Advanced to Challenge, a step which some dancers (perhaps due to their learning style) never really master. If we are talking about square dancing, including Mainstream and C4, I have to include Phantoms.
Finally, Rewind is the canonical example of something that was considered and rejected by the majority of C4 dancers as being too hard, and "this is not what we want in our dancing". It must be included in order to show the limits of square dancing (or "not square dancing").